Imithetho kaKepler

Ngaphaya konyaka we-1600, uJohannes Kepler wachitha iminyaka ezama ukwenza iiplanethi zihambe ngemijikelezo egqibeleleyo. Ukujonga kweMars kwayeka ukufakela. Into ayifumene endaweni yoko yimithetho emithathu esiya kuqhuba yonke inqanawa yehlabathi esiyikhuphayo.

Amanani akazange alungele izijikelezi. Ke ke uKepler watshintsha iitafile.

Bona

Ngaphaya konyaka we-1600, uJohannes Kepler wasebenza ngeminyaka eninzi yokujonga ngokuchanekileyo iMars eyayenziwe nguTycho Brahe. Wonke umntu wacinga ukuba iiplanethi zihamba ngemijikelezo egqibeleleyo. Amanani akayi kudibana. Emva kweminyaka yomsebenzi, uKepler wafumana imithetho emithathu elungele.

1. Ii-orbits zii-ellipsis (izijikelezo ezithe nkqo), ngelanga elincinci elingumbindi kwinto ebizwa ngokuba yi focus.
2. Iplanethi ihamba ngokukhawuleza phezulu kufutshane nelanga kwaye ihamba phantsi ngokucothayo kude nalo. Zoba umgca ukusuka kwilanga ukuya kwiplanethi: isusa iindawo ezilinganayo kwiixesha ezilinganayo.
3. Ii-orbit ezikhulu zithatha ixesha elide kakhulu. Ayinakuba likhulu kuphela kuba indlela ibanzi, kodwa inkulu ngokwenani.

Kwibhokisi yesand ephezulu, iHalley’s Comet ilayishelwe. Bona ukuba ijikeleza ilanga kwaye emva koko ihamba ngeenyawo kwinxalenye yangaphandle yendlela yayo. Iinxalenye ezibomvu zitywinwa zonke kwixesha elifanayo: incinci kwaye ide kude, incinane kwaye ibanzi phantsi. Indawo efanayo, ixesha elinye. Lo ngumthetho wesibini osebenza.

Ukuqonda

Ii-ellipsis zichazwa ngamaxabiso amabini:

• Isiqingatha-sokuqala se-axis a: iphesenti yobubanzi obude kakhulu. Imisela ubungakanani.
• I-eccentricity e: ukuba ijikeleze njani. 0 yidayari; ifutshane ne-1 inde kakhulu kwaye incinci.

Incopho efutshane nelanga yi periapsis (i-perihelion yelanga), kumgama a( 1 - e). Incopho ekufuphi kakhulu yi apoapsis (i-aphelion), kwi a(1 + e).

Injongo	a	e
Ihlabathi	1.000	0.017
I-Mars	1.524	0.093
I-Halley’ s Comet	17.9	0.967

Imithetho yesithathu. Kuzo zonke izinto ezijikeleza ilanga, isikwere sexesha silingana ne cube ye-semi- major axis. kwiminyaka ne

T² = a³

IJupiter: a = 5.20 au, ngoko T = 5.20^1.5 = 11.9 iminyaka. IHalley: a = 17.9 au inika T = 75.9 iminyaka, ecaleni kwembali yayo yeminyaka eyi-75 ukuya kwi-76 yokubuyela.

Ibhokisi yesand ibonisa T²/a³ kwiminyaka² nganye yeau³. Tshintsha i orbit ejikeleze i Sun ngokwaneleyo njengoko ufuna: iyahlala 1.

Umphathi

UNewton wazisa zonke izivumelwano ezintathu ukusuka kwisivumelwano sakhe soxinzelelo. Umthetho wesithathu ngefom epheleleyo yi:

T = 2π · √(a³ / μ),   μ = G(M + m)

Ngoko ke T²/a³ = 4π²/μ ifana neyodwa kwi orbit nganye ejikeleze isiqu esisisiseko, kodwa ihlukile kwiziqu ezahlukeneyo. Guqula isiqu sebhokisi yethayile esiphakathi kumhlaba kwaye ixabiso elihlala likhona liyi malunga ne-9.9 × 10⁻⁵ s² km⁻³ endaweni ye-1 yr² au⁻³. I-m kwi-μ yiyo loo nto uguqulelo lweKepler luncinci olungahambi kakuhle ngeJupiter, eyona nto ininzi ngayo malunga ne-0.1% yelanga.

Umthetho wesibini ukhuselo lwe momentum ye-angle. Uxinzelelo lwendawo yi dA/dt = h/2, apho h = r²·dν/dt lixesha elikhethekileyo le-engile, elihlala likhona nakweyiphi na intshukumo esembindini. Kwindawo ejikelezayo neyokuphuma isantya sihamba ngokuthe nkqo kwi-radius, ngoko r_p·v_p = r_a·v_a: uthelekiso lwesantya lulingana nothelekiso lomgama oguqukayo. KwiHalley, r_a/r_p = (1 + e) /(1 − e) ≈ 60, ngoko ihamba malunga nexesha le-60 likhawulezayo kwindawo ephakathi kwelanga nelanga elijikelezayo kunendawo engaphandle kwelanga.

Ukubeka iqulethe kwindlela yayo yokujikeleza ngexesha elinikezelweyo ufuna umlinganiselo wokungalingani M = n (t − t_ p), ngemilinganiselo yentshukumo n = 2π/ T, kwaye ngoko uthelekiso lweKepler, M = E − e sin E. Akukho sizathu esivaliweyo; izixhobo kule ndawo zisombulula ngeNewton iteration, edibanisa kwinyathelo elincinci nakwiHalley’s e=0.967.